Calculating Volume and Side Length of a Cut Cone - Is This Right

[Ryoukomaru]

This is a problem I had in a test and almost everyone got different answers for it, we discussed and well, I spotted mistakes in their solutions so I think mine is right but I wanted to check here and also ask if there is an easier/faster way to do it.

There is a container that is similar to the bottom part of a cone which is cut into half. Top radius is $$10cm$$ and bottom is $$20cm$$. And the volume of this container is $$500cm^{3}$$. What is the length of the slanted side ?

So what I did was, first I drew this. ;P (See attachment)

Then what I did was to write an equation for $$V_2$$ in terms of $$V_1$$ and $$V_T$$

By solving the equation for height, I got $$h=1.5915$$
Then I used Pythagoras' theorem to find the length of the slanted side and it comes to:
$$10^2+2h^2=s^2=>100+3.183^2=s^2=> s=10.4944$$

Is this correct ? And is there a formula to find the volume of this shape ?

 

[Tide]

The volume of the "flower pot" section is

$$V = \frac {\pi h}{3} (R^2 + rR + r^2)$$

where R is the large radius, r is the small radius and h is the same as your x. It's simply the difference between the volume of the large cone and the small cone.